Wentworth's Plane and Solid Geometry · Book VI, §§ 536–548 (Prop. XVI) · Book 7: Book VI: Lines and Planes in Space
86. Dihedral angles
Goal: Your student can define a dihedral angle, its edge, faces and plane angle, and prove that dihedral angles with equal plane angles are equal.
You'll need
- The book, open to these pages
- A ruler (straightedge) and a pair of compasses
- Sharp pencil, eraser and plain paper (or a geometry notebook)
- Two or three stiff cards (to stand for planes) and a few pencils or straws (to stand for lines)
What's in the book
Definitions of dihedral angles (§§ 536–546), the list of properties that carry over from plane angles (§ 547), and Proposition XVI.
The lesson, step by step
Warm up
Ask: What is an angle in plane geometry? How do we measure it?
The new idea
The opening between two intersecting planes is called a dihedral angle.
The plane angle of a dihedral angle is the plane angle formed by two straight lines, one in each plane, perpendicular to the edge at the same point.
Two dihedral angles are equal if their plane angles are equal.
Draw the figure
Open a book partway and stand it on the table: the two covers are the faces, the spine is the edge. Hold a protractor (or a folded card) against the spine at right angles to it to see the plane angle.
Draw a dihedral angle M-AB-N and mark its plane angle at one point of the edge.
Walk through the proof
§ 546: the plane angle is the same wherever on the edge you draw it, since the sides at two points are parallel (§ 534).
Prop. XVI: lay one plane angle on the other. The edge, being perpendicular to the plane of the angle, must fall along the other edge (only one perpendicular, § 511). Two planes with two intersecting lines in common coincide (§ 497). So the faces coincide.
Your student proves it back
Close the book. Your student sketches the figure from memory (a simple perspective drawing is fine), letters it the same way, and gives the proof aloud: what is given, what is to be proved, then each step with its reason ("by § 501", "by § 507"). Point to each line and plane as it is named.
If a step is missing, ask "How do we know that?" rather than supplying it.
Exercises
Read the six properties in § 547 aloud. For each, ask your student to name the plane-geometry theorem it copies (for example, 3 copies "vertical angles are equal").
Wrap up
Ask: What is the dihedral angle between a wall and the floor? Between an open door and its wall?